It is well known that there is inter-channel redundancy inside convolution filters, which results in the flexibility problem in convolution operation [1]}. Compared to the standard convolution, the involution kernel \(\mathbf {\mathcal {H}}\in \mathbb {R}^{H\times W\times K\times K\times G}\)  is designed to realize transforms with inverse characteristics in the spatial and channel domain. Specifically, an involution kernel \(\mathbf {\mathcal {H}}_{i,j,\cdot ,\cdot ,g}\in \mathbb {R}^{K\times K}, g=1,2,\cdots ,G\) , is specially adapted for the pixel \(\mathbf {X}_{i,j}\in \mathcal {R}^C\)  located at the corresponding coordinate \((i, j)\) , but shared over the channels. \(G\)  denotes the number of groups and each group shares the same involution kernel. The output feature map of involution is derived by executing multiply-add operations on the input with the involution kernels, given as
\(\mathbf {Y}_{i,j,k}=\sum _{(u,v)\in }\mathbf {\mathcal {H}}_{i,j,u+\lfloor K/2\rfloor ,v+\lfloor K/2\rfloor ,\lceil kG/C\rceil }\mathbf {X}_{i+u,j+v,k}.\) 
